COMPACT CLIFFORD-KLEIN FORMS OF HOMOGENEOUS SPACES OF SO(2,n) HEE OH AND DAVE WITTE Abstract. A homogeneous space G/H is said to have a compact Clifford-Klein form if there exists a discrete subgroup Γ of G that acts properly discontinuously on G/H, such that the quotient space Γ\G/H is compact. When n is even, we find every closed, connected subgroup H of G = SO(2, n), such that G/H has a compact Clifford-Klein form, but our classification is not quite complete when n is odd. The work reveals new examples of homogeneous spaces of SO(2, n) that have compact Clifford-Klein forms, if n is even. Furthermore, we show that if H is a closed, connected subgroup of G = SL(3, R), and neither H nor G/H is compact, then G/H does not have a compact Clifford-Klein form, and we also study noncompact Clifford-Klein forms of finite volume. 1. Introduction 1.1. Assumption. Throughout this paper, G is a Zariski-connected, almost simple, linear Lie group (“Almost simple” means that every proper normal subgroup of G either is finite or has finite index.) In the main results, G is assumed to be SO(2,n) (with n 3). There would be no essential loss of generality if one were to require≥ G to be connected, instead of only Zariski connected (see 3.11). However, SO(2,n) is not connected (it has two components) and the authors prefer to state results for SO(2,n), instead of for the identity component of SO(2,n). 1.2. Definition. Let H be a closed, connected subgroup of G. We say that the homogeneous space G/H has a compact Clifford-Klein form if there is a discrete subgroup Γ of G, such that Γ acts properly on G/H; and • Γ G/H is compact. • \ (Alternative terminology would be to say that G/H has a tessellation, because the Γ-translates of a fundamental domain for Γ G/H tessellate G/H, or one could simply say that G/H has a compact quotient.) See the surveys\ [Kb5] and [Lab] for references to some of the previous work on the existence of compact Clifford-Klein forms. We determine exactly which homogeneous spaces of SO(2,n) have a compact Clifford-Klein form in the case where n is even (see 1.7), and we have almost complete results in the case where n is odd (see 1.9). (We only consider homogeneous spaces G/H in which H is connected.) The work leads to new examples of homogeneous spaces that have compact Clifford-Klein forms, if n is even (see 1.5). We also show that only the obvious homogeneous spaces of SL(3, R) have compact Clifford-Klein forms (see 1.10), and we study noncompact Clifford-Klein forms of finite volume (see 6). § 1.3. Notation. We realize SO(2,n) as isometries of the indefinite form v v = v1vn+2 + v2vn+1 + n v2 on Rn+2 (for v = (v , v ,...,v ) Rn+2). Let A be theh subgroup| i consisting of the i=3 i 1 2 n+2 ∈ Pdiagonal matrices in SO(2,n) whose diagonal entries are all positive, and let N be the subgroup consisting of the upper-triangular matrices in SO(2,n) with only 1’s on the diagonal. Thus, the Date: March 3, 1999 (Corrected version). 1 2 HEE OH AND DAVE WITTE Lie algebra of AN is t1 φ x η 0 t2 y 0 η − t1,t2, φ, η R, (1.4) a + n = 0 yT xT . Rn∈−2 − − x,y t2 φ ∈ − − t1 − Note that the first two rows of any element of a + n are sufficient to determine the entire matrix. Let us recall a construction of compact Clifford-Klein forms found by Kulkarni [Kul, Thm. 6.1] (see also [Kb1, Prop. 4.9]). The subgroup SU(1,m), embedded into SO(2, 2m) in a standard way, acts properly and transitively on the homogeneous space SO(2, 2m)/ SO(1, 2m). Therefore, any co-compact lattice Γ in SO(1, 2m) acts properly on SO(2, 2m)/ SU(1,m), and the quotient Γ SO(2, 2m)/ SU(1,m) is compact. Now let H = SU(1,m) (AN). Then the Clifford-Klein \ SU ∩ form Γ SO(2, 2m)/HSU is also compact, since SU(1,m)/HSU is compact. (Similarly, Kulkarni also constructed\ compact Clifford-Klein forms Λ SO(2, 2m)/ SO(1, 2m), by letting Λ be a co-compact lattice in SU(1,m).) \ The following theorem demonstrates how to construct new examples of compact Clifford-Klein forms Γ SO(2, 2m)/HB . The subgroup HB of SO(2, 2m) is obtained by deforming HSU, but HB is almost\ never contained in any conjugate of SU(1,m). 1.5. Theorem. Assume that G = SO(2, 2m). Let B : R2m−2 R2m−2 be a linear transformation that has no real eigenvalue. Set → t 0 x η 0 x R2m−2, (1.6) hB = t B(x) 0 η ∈ , − t, η R . ∈ let HB be the corresponding closed, connected subgroup of G, and let Γbe a co-compact lattice in SO(1, 2m). Then 1) the subgroup Γ acts properly on SO(2, 2m)/HB ; 2) the quotient Γ SO(2, 2m)/H is compact; and \ B 3) HB is conjugate via O(2, 2m) to a subgroup of SU(1,m) if and only if for some a, b R (with b = 0), the matrix of B with respect to some orthonormal basis of R2m−2 is a block∈ 6 a b diagonal matrix each of whose blocks is . b a − Furthermore, by varying B, one can obtain uncountably many pairwise nonconjugate subgroups. We recall that T. Kobayashi [Kb6, Thm. B] showed that a co-compact lattice in SU(1,m) can be deformed to a discrete subgroup Λ, such that Λ acts properly on SO(2, 2m)/ SO(1, 2m) and the quotient space Λ SO(2, 2m)/ SO(1, 2m) is compact, but Λ is not contained in any conjugate of SU(1,m). (This example\ is part of an extension of work of W. Goldman [Gol].) Note that Kobayashi created new compact Clifford-Klein forms by deforming the discrete group while keeping the homogeneous space SO(2, 2m)/ SO(1, 2m) fixed. In contrast, we deform the homogeneous space SO(2, 2m)/HSU to another homogeneous space SO(2, 2m)/HB while keeping the discrete group Γ in SO(1, 2m) fixed. For even n, we show that the Kulkarni examples and our deformations are essentially the only interesting homogeneous spaces of SO(2,n) that have compact Clifford-Klein forms. We assume that H AN, because the general case reduces to this (see 3.9). ⊂ 1.7. Theorem (cf. Thm. 5.7). Assume that G = SO(2, 2m). Let H be a closed, connected subgroup of AN, such that neither H nor G/H is compact. The homogeneous space G/H has a compact Clifford-Klein form if and only if either CLIFFORD-KLEIN FORMS OF SO(2,n)/H 3 1) H is conjugate to a co-compact subgroup of SO(1, 2m); or 2) H is conjugate to HB, for some B, as described in Theorem 1.5. It is conjectured [Kb6, 1.4] that if H is reductive and G/H has a compact Clifford-Klein form, then there exists a reductive subgroup L of G, such that L acts properly on G/H, and the double- coset space L G/H is compact. Because there is no such subgroup L in the case where G = SO(2, 2m+1)\ and H = SU(1,m) (see 5.10), the following is a special case of the general conjecture. 1.8. Conjecture. For m 1, the homogeneous space SO(2, 2m + 1)/ SU(1,m) does not have a compact Clifford-Klein form.≥ If this conjecture is true, then, for odd n, there is no interesting example of a homogeneous space of SO(2,n) that has a compact Clifford-Klein form. 1.9. Theorem. Assume that G = SO(2, 2m + 1), and that G/ SU(1,m) does not have a compact Clifford-Klein form. If H is a closed, connected subgroup of G, such that neither H nor G/H is compact, then G/H does not have a compact Clifford-Klein form. The main results of [OW] list the homogeneous spaces of SO(2,n) that admit a proper action of a noncompact subgroup of SO(2,n) (see 2). Our proofs of Theorems 1.7 and 1.9 consist of case- by-case analysis to decide whether each of§ these homogeneous spaces has a compact Clifford-Klein form. The following proposition does not require such a detailed analysis, but is obtained easily by combining theorems of Y. Benoist (see 7.1) and G. A. Margulis (see 3.6). 1.10. Proposition. Let H be a closed, connected subgroup of G = SL(3, R). If G/H has a compact Clifford-Klein form, then H is either compact or co-compact. The paper is organized as follows. Section 2 recalls some definitions and results, mostly from [OW]. Section 3 presents some general results on Clifford-Klein forms. Section 4 proves Theo- rem 1.5, the new examples of compact Clifford-Klein forms. Section 5 proves Theorems 1.7 and 1.9, the classification of compact Clifford-Klein forms. Section 6 discusses noncompact Clifford-Klein forms of finite volume.